Artificial intelligence (AI) holds significant promise for advancing natural disaster management through the use of predictive models that analyze extensive datasets, identify patterns, and forecast potential disasters. These models facilitate proactive measures such as early warning systems (EWSs), evacuation planning, and resource allocation, addressing the substantial challenges associated with natural disasters. This study offers a comprehensive exploration of trustworthy AI applications in natural disasters, encompassing disaster management, risk assessment, and disaster prediction. This research is underpinned by an extensive review of reputable sources, including Science Direct (SD), Scopus, IEEE Xplore (IEEE), and Web of Science (WoS). Three queries were formulated to retrieve 981 papers from the earliest documented scientific production until February 2024. After meticulous screening, deduplication, and application of the inclusion and exclusion criteria, 108 studies were included in the quantitative synthesis. This study provides a specific taxonomy of AI applications in natural disasters and explores the motivations, challenges, recommendations, and limitations of recent advancements. It also offers an overview of recent techniques and developments in disaster management using explainable artificial intelligence (XAI), data fusion, data mining, machine learning (ML), deep learning (DL), fuzzy logic, and multicriteria decision-making (MCDM). This systematic contribution addresses seven open issues and provides critical solutions through essential insights, laying the groundwork for various future works in trustworthiness AI-based natural disaster management. Despite the potential benefits, challenges persist in the application of AI to natural disaster management. In these contexts, this study identifies several unused and used areas in natural disaster-based AI theory, collects the disaster datasets, ML, and DL techniques, and offers a valuable XAI approach to unravel the complex relationships and dynamics involved and the utilization of data fusion techniques in decision-making processes related to natural disasters. Finally, the study extensively analyzed ethical considerations, bias, and consequences in natural disaster-based AI.